Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 78

Table 6 Example of price uncertainty
E t ðS T Þ RPð 0 2 Þ l l r Market situation

;

105.00 − 5.00 0.0544 0.0244 Normal backwardation

100.00 0.00 0.0300 0.0000

-

95.00 5.00 0.0044 0.0256 Contango

−

These values correspond to a different discount rate in the real world. For
example for RP

105e 2 l ¼ 5 00 it must hold that 94 176 ¼ In general:

: : :

1
T t ln

F ð t ; T Þ
E t ðS T Þ

l ¼ ð 19 Þ

As can be seen in the example, when E t ðS T Þ ¼ 105 00 the discount rate with risk

:

is l ¼ 5 44 % . However it is very hard to estimate E t ðS T Þ ; while Fðt ; TÞ is deduced

:

is certain, 14 the
corresponding amount can be discounted at the riskless interest rate r and
the resulting value should be the spot price. The method used is valid assuming that
the market is complete.
Now assume that the spot price is S t ¼

directly from the market quotation. Since the future price Fðt ; TÞ

97 00 It is known that:

: :

Fðt ; TÞ ¼ S t e ðrdÞðTtÞ ð 20 Þ

Therefore:

¼

1
T t ln

F ð t ; T Þ
S t

d ¼ r 0 01477 ð 21 Þ

:

The convenience yield may not be constant, and may vary over time.

(B) Annuities ( GBM case )

In this second example, the aim is to deduce the value of an annuity when the
price follows a GBM process, e.g., the price of CO 2 emission allowance over
20 years.
Based on the future equation:

; tÞ ¼ S 0 e ðakÞt

Fð ð 22 Þ

0

An annuity between s 1 and s 2 has a value of:

Z

s 2

S 0
a k r ½

S 0 e ðakÞt e rt d t ¼

e ðakrÞs 2 e ðakrÞs 1

Vðs 1 ; s 2 Þ ¼ ð 23 Þ

s 1

14

The markets perform the role of covering the counterparty risk.